[Paper Review] Schubert calculus and torsion explosion

Geordie Williamson|arXiv (Cornell University)|Sep 19, 2013

Advanced Algebra and Geometry36 references28 citations

TL;DR

This paper demonstrates that certain Schubert calculus numbers in SLₙ, which grow exponentially, correspond to torsion in Soergel bimodules and thus imply the failure of Lusztig’s conjecture on characters of simple modules in positive characteristic when p ≤ h. The authors use intersection forms and generators/relations of Soergel bimodules to detect p-torsion, providing counterexamples to both Lusztig’s and James’ conjectures in higher rank.

ABSTRACT

We observe that certain numbers occurring in Schubert calculus for SL_n also occur as entries in intersection forms controlling decompositions of Soergel bimodules and parity sheaves in higher rank. These numbers grow exponentially. This observation gives many counterexamples to Lusztig's conjecture on the characters of simple rational modules for SL_n over a field of positive characteristic. We explain why our examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.

Motivation & Objective

• To investigate the connection between Schubert calculus in SLₙ and torsion in Soergel bimodules.
• To test the validity of Lusztig’s conjecture on characters of simple modules in positive characteristic when p ≤ h.
• To provide counterexamples to Lusztig’s conjecture and the James conjecture on decomposition numbers for symmetric groups.
• To establish that torsion in intersection cohomology stalks can be detected via intersection forms in Soergel bimodule theory.
• To demonstrate that exponential growth in Schubert calculus numbers leads to torsion explosion in higher-rank settings.

Proposed method

• The authors use generators and relations for the monoidal category of Soergel bimodules to compute intersection forms for pairs (w, x), where w ∈ W and x ≤ w.
• They analyze the elementary divisors of these integral intersection forms to detect p-torsion in stalks and costalks of integral intersection cohomology complexes.
• The method relies on the equivalence between absence of p-torsion and the truth of Lusztig’s conjecture for p > h, via parity sheaves and Soergel’s categorification.
• They apply this to the subquotient category around the Steinberg weight, which models the principal block of rational representations.
• The authors use the theory of moment graphs and the work of Elias and Williamson to simplify torsion detection compared to traditional topological methods.
• They leverage known results on torsion in flag varieties (e.g., Braden, Polo) and extend them via algebraic computation in Soergel bimodule categories.

Experimental results

Research questions

• RQ1 Do Schubert calculus numbers in SLₙ correspond to torsion in Soergel bimodules, and if so, how does this affect Lusztig’s conjecture?
• RQ2 Can exponential growth in Schubert calculus numbers lead to torsion explosion in higher-rank Lie groups?
• RQ3 Does the failure of Lusztig’s conjecture occur when p ≤ h, and can this be detected via intersection forms?
• RQ4 Can the James conjecture on decomposition numbers for symmetric groups be invalidated by torsion in Soergel bimodules?
• RQ5 Is there a systematic algebraic method to detect p-torsion in intersection cohomology using Soergel bimodules?

Key findings

• The paper identifies Schubert calculus numbers in SLₙ that grow exponentially and correspond to torsion in Soergel bimodules.
• These numbers lead to counterexamples to Lusztig’s conjecture on characters of simple modules when p ≤ h.
• The authors provide explicit counterexamples to the James conjecture on decomposition numbers for symmetric groups.
• Torsion in intersection cohomology stalks is detected via elementary divisors of intersection forms in Soergel bimodule categories.
• The method using generators and relations for Soergel bimodules provides a computationally feasible way to detect p-torsion, avoiding complex topological computations.
• The results show that torsion is not limited to low-rank cases, as previously thought, and can occur in high-rank flag varieties such as A₄ₙ₋₁ and E₆.